The Entity Modelling Tutorial Part One
Entities, Relationships and Attributes

Information Structure Choices

In this section we discuss various ways of representing directed graphs, the different
shapes that their entity models can take and the different organisations of information
implied by the different shapes.

Directed Graphs

There is a reliance here, in this presentation, on the use of the work ‘abstract’
and
on an understanding of what the mathematician means by this.
The best that I can offer as explanation is that the mathematical idea of an abstract
structure
is that of a set of individuals and relations independent of any particular
representation of the individuals and the mathematical concepts of graph and directed graph are cases in point.
In the case of a graph the individuals are notionally
vertices and edges and the relations are the
incidence relations between them. It is certainly ironic and perhaps at first sight
paradoxical, but with regard to any particular
abstract structure, to describe it we need some kind
of representation even though what we seek to describe is something without any particular representation. Mathematican Robin Gandy describes this to a non-mathematical
audience in
a lecture ‘Structure in Mathematics’
1; he draws the love
relationships to be found in Iris Murdoch's novel Severed Head as a
directed graph and I have drawn an equivalent graph in figure 33.
If we abstract a directed graph from this then there are no labels, no lines,
no points on paper just the facts of a set of notional individual entities and ther
relationships between them2.

pLh

hLp

aLp

pLa

pLg

mLp

mLh

aLx

xLa

gLx

gLm

mLg

Table

Gandy's relational depiction of the same love relationships.1

Figure 33

Example from ‘Structure in Mathematics’ of a directed graph showing
the love relationships to be found in Iris Murdoch's novel ‘Severed Head’.
The labels on the graph are simply to show the correspondence with the novel.

As an illustration of different representations of the same abstract structure,
Gandy gives the relational representation shown in table .
In the relational representation,
p occupies the same structural position as
PALMER
,
h as
HONOR
,
a as
ANTONIA
,
g as
GEORGIE
,
m as
MARTIN
,
x as
ALEXANDER
.
This relational description can be described by the message structure:

table
=>

loves*

loves
=>

lover,

loved

Gandy's depicted love relationship is a many-many relationship and we can represent
in an entity model like this:

Equivalently in the technical vocabulary but with the same shape we have the entity
model
of a directed graph shown in figure 34

Figure 34

The model of a directed graph.

There is of course nothing special about this
example —
loves is a recursive many-many
relationship and
any such could be used to illustrate that a recursive many-many relationship is structurally
a directed graph.

Gandy's relational representation shown in figure 33 could be rearranged in
two ways — from the point of view of the lover or the loved; these are shown in figures
35
and 36.

aLp,x

gLx,m,g

hLp

mLp,h,g

pLh,a,g

xLa

Figure 35

This representation can be described by the message structure:

table
=>

loves*

loves
=>

lover,

loved*

From this representation you may quickly find who is loved by a, who is loved by g and so on.

x,pLa

m,pLg

m,pLh

gLm

h,mLp

a,gLx

Figure 36

This representation is described by the message structure:

table
=>

loves*

loves
=>

lover*,

loved

From this representation you may quickly determine who loves a, who loves g and so on.

ER Models of Directed Graph

Viewing figures 36 and following on from figure 36 we see that there are two
further ways of modelling directed graphs. These are shown in figure 37 and 38.

Figure 37

Directed Graph ER Model (ii)

Figure 38

Directed Graph ER Model (iii)

Though we have given three ways of modelling directed graphs note that mathematically
there is only a single concept.
The different models represent different ways of localising and communicating the
information content of a graph.
The models in figures 37 and 38 vary the incidence relationships between edges and
vertices between
categories reference and composition. If both are made composition then we get a further variation which is shown in
figure 39.

Figure 39

Directed Graph ER Model (iv)

But what communication structure do we associate with this? Well, this way of modelling
directed graphs is akin to the matrix structure modelled in chapter one.
Applied to the representation of the Severed Head love relationships it implies a
communication in which
it is apparent both (i) who loves each subject a,g,h and so on and (ii) who each subject a,g,h, etc.
is loved by. This is made apparent by a matrix representation:

a

g

h

m

p

x

a

x

x

g

x

x

h

x

x

m

p

x

x

x

x

x

x

x

We mentioned in chapter one that information is usually hierarchically communicated
whereas with double composition relationships there is not one hierarchy but two hierarchies.
Information must
be double communicated once in each hierarchical form.
For linear, hierarchical communication the above model is
therefore replaced by one in which both hierarchies are represented as shown in figure
40.

Figure 40

Directed Graph ER Model (v)

Modelling Graphs and Symmetric Relationships

We use the unqualified term graph to mean a set of vertices and a set of undirected
connections
or edgess between them. To the mathematician the definition is straightforward but to the
entity
modeller it is less so and is somewhat unsatisfying.
There is a difficulty and this difficulty lies at the heart of entity modelling.
An understanding of the difficulty reveals simultaneously a gap in the entity relational
approach by comparison with the mathematical approach but also an affinity between
it and the means of language and communication.

To model a bidirectional graph is to add a reverse relationship to any of the models
of a directed graph.
If we start from figure 37 then we get this model:

In this model we have greyed out the to relationship to show that it has become redundant.
This is because the vertex that an exit leads to is always the same vertex that its
reverse leads from.

2
According to Gandy: This indifference to form of
representation
is what Bertrand Russell referred to when he said that in mathematics we are not interested
in
what we are talking about. We return to the subject abstraction versus representation
in a later section.